Let a line L : 2x + y = k, k 0 be a tangent to the hyperbola x2 - y2 = 3.

Question

Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 - y2 = 3. If L is also a tangent to the parabola y2 = $$\alpha$$x, then $$\alpha$$ is equal to :

Accepted Answer

Tangent to hyperbola of

Slope m = -2 (given)

y = -2x $$\pm$$ $$\sqrt {3(3)} $$

$$\left( {y = mx \pm \sqrt {{a^2}{m^2} - {b^2}} } \right)$$

$$\Rightarrow$$ y + 2x = $$\pm$$ 3 $$\Rightarrow$$ 2x + y = 3 (k > 0)

For parabola y2 = $$\alpha$$x

$$y = mx + {\alpha  \over {4m}}$$

$$ \Rightarrow y =  - 2x + {\alpha  \over { - 8}}$$

$$ \Rightarrow {\alpha  \over { - 8}} = 3$$

$$ \Rightarrow \alpha  =  - 24$$

Solutions · 1